That's ok Mathwonk. I figured you are busy. Thanks for the Walker reference and the other references above. They're helpful. Also, I'm afraid I used an abuse of notation above. That's really three factors:
[tex]f(z,w)=\left(wg(z)\right)\left(wh(\epsilon_1z^{1/2})\right)\left(wh(\epsilon_2 z^{1/2})\right)[/tex]
however the two factors of [itex]h(z^{1/2})[/itex] come from the same 2valued manifold, just different singlevalued determinations of it. Just thought my condenced notation was cleanerlooking.
And the expansion comes from NewtonPuiseux's Theorem:
The quotient field [itex]\mathbb{C}((z^*))[/itex] of convergent fractional power series (which I assume includes all analytic functions) is algebraically closed so for [itex]f(z,w)=a_n(z)w^n+\cdots+a_0(z)[/itex], with [itex]a_i(z)\in \mathbb{C}((z^*))[/itex], we have
[tex]f(z,w)=\prod_{i=1}^n \left(wg_i(z^{1/k_i})\right)[/tex]
with [itex]g_i(z^{1/k_i})\in\mathbb{C}((z^*))[/itex]. One reason I knew the expansion was in terms of [itex]g(z)[/itex] and [itex]h(z^{1/2})[/itex], is because I drew it:
(Not entirely sure I'm quoting that theorem precisely. This is all very new to me.)
